example the Leray spectral sequence. An injective sheaf F {\displaystyle {\mathcal {F}}} is a sheaf that is an injective object of the category of abelian sheaves;...

enough injectives; that is, for every sheaf E there is an injective sheaf I with an injection E → I. It follows that every sheaf E has an injective resolution:...

does not imply that all the restriction maps of this sheaf are injective!) In contrast, for the sheaf of smooth functions on a smooth manifold, germs contain...

Look up sheaf in Wiktionary, the free dictionary. In mathematics, a sheaf (pl.: sheaves) is a tool for systematically tracking data (such as sets, abelian...

In mathematics, a sheaf of O-modules or simply an O-module over a ringed space (X, O) is a sheaf F such that, for any open subset U of X, F(U) is an O(U)-module...

Γ(U, A)/Γ(U, J) → Γ(U, A/J) for open subsets U is injective, but not surjective in general. (See sheaf cohomology.) In the context of schemes, the importance...

notion was introduced by A. Joyal in the 1970s. Similarly, a simplicial sheaf on a site is a simplicial object in the category of sheaves on the site...

I^{0}\to I^{1}\to I^{2}\to \cdots } where the I i are all injective (this is known as an injective resolution of X). Applying the functor F to this sequence...

Central extension Splitting lemma Projective module Injective module Projective resolution Injective resolution Koszul complex Exact functor Derived functor...

flat modules. Similarly every module has injective resolutions, which are right resolutions consisting of injective modules. Given a module M over a ring...

comes from. The classical definition of a sheaf begins with a topological space X {\displaystyle X} . A sheaf associates information to the open sets of...

it is a fractional ideal sheaf (see below). Conversely, every rank one reflexive sheaf corresponds to a Weil divisor: The sheaf can be restricted to the...

group scheme G on a scheme X over a base scheme S, an equivariant sheaf F on X is a sheaf of O X {\displaystyle {\mathcal {O}}_{X}} -modules together with...

enough injectives. Embedding X into some injective object I0, the cokernel of this map into some injective I1 etc., one constructs an injective resolution...

{\displaystyle X/G} is G {\displaystyle G} -invariant. Also, the pullback is an injective morphism. In our case of R n / Z n {\displaystyle \mathbb {R} ^{n}/\mathbb...

computation of sheaf cohomology using an acyclic resolution of the sheaf in question. Let F {\displaystyle {\mathcal {F}}} be a sheaf on a topological...

a sheaf for any Grothendieck topology on a category in a similar way. The category of sheaves of abelian groups over a scheme has enough injective objects...

a supermanifold M is contained in its sheaf OM of "smooth functions". In the dual point of view, an injective map corresponds to a surjection of sheaves...

is a contravariant left-exact functor; it is exact if and only if A is injective. If k is a field and V is a vector space over k, we write V * = Homk(V...

Abelian categories having enough injectives, F {\displaystyle F} a left-exact functor, and G {\displaystyle G} sending injective objects to F {\displaystyle...

field analogue of complex multiplication theory. A shtuka (also called F-sheaf or chtouca) is a sort of generalization of a Drinfeld module, consisting...

lemma: Lemma — If K is an injective complex in an abelian category C such that the kernels of the differentials are injective objects, then for each n...

consists only of its neutral element. The monomorphisms in Ab are the injective group homomorphisms, the epimorphisms are the surjective group homomorphisms...

topological spaces, monomorphisms are precisely the injective continuous functions; but not all injective continuous functions are subspace embeddings. In...

scheme, Čech and sheaf cohomology agree for any quasi-coherent sheaf. For the étale topology, the two cohomologies agree for any étale sheaf on X, provided...

Similarly, if F {\displaystyle {\mathcal {F}}} is a sheaf on X, then there is a corresponding sheaf F an {\displaystyle {\mathcal {F}}^{\text{an}}} on...

the sheaf of relative differentials is stably isomorphic to an ( n + 1 ) {\displaystyle (n+1)} -fold sum of the dual of the Serre twisting sheaf. The...

Mehta and Ramanathan (1985), is a splitting of the injective morphism OX→F*OX from a structure sheaf OX of a characteristic p > 0 variety X to its image...

map (respectively, a continuous map, etc.). The restriction of the non-injective function f : R → R ,   x ↦ x 2 {\displaystyle f:\mathbb {R} \to \mathbb...

non-negative number: it has a "fork" at zero. The etale space of a sheaf, such as the sheaf of continuous real functions over a manifold, is a manifold that...